Allen-Cahn Equation

This section repeats the adaptive PINN method presented by Wight and Zhao.

The Allen-Cahn equation has the following general form:

$$ \partial_{t} u=\gamma_{1} \Delta u+\gamma_{2}\left(u-u^{3}\right). $$

Consider the one-dimensional Allen-Cahn equation with periodic boundary conditions:

$$ \begin{array}{l} u_{t}-0.0001 u_{x x}+5 u^{3}-5 u=0, \quad x \in[-1,1], \quad t \in[0,1], \ u(0, x)=x^{2} \cos (\pi x) \ u(t,-1)=u(t, 1) \ u_{x}(t,-1)=u_{x}(t, 1). \end{array} $$

Periodic Boundary Conditions

The periodic boundary conditions are enforced by $u(t, x)=u(t,x+2)$ and $u_x(t, x)=u_x(t,x+2)$ with $x=-1$, which is equivalent to

$$ \begin{array}{l} \tilde u(t,x)=u(t,x+2), \quad \forall t\in[0,1],x\in[-1,1], \ \tilde u(t,x)=u(t,x),\quad \forall t\in[0,1],x=-1, \ \tilde u_x(t,x)=u_x(t,x),\quad \forall t\in[0,1],x=-1.\ \end{array} $$

The transform above is implemented by

net_u = sc.MLP([2, 128, 128, 128, 128, 2], activation=sc.Activation.tanh)
net_u = sc.NetNode(inputs=('x', 't',), outputs=('u',), name='net1', net=net_u)
xp = sc.ExpressionNode(name='xp', expression=x + 2)
net_tilde_u = sc.get_shared_net_node(net_u, inputs=('xp', 't',), outputs=('up',), name='net2', arch='mlp')

where xp translates $x$ to $x+2$. The node net_tilde_u has the same internal parameters as net_u while its inputs and outputs are translated.

Receivers acting as Callbacks

We define a group of Signal to trigger receivers. They are adequate for customizing various PINN algorithms at the moment.

class Signal(Enum):
    REGISTER = 'signal_register'
    SOLVE_START = 'signal_solve_start'
    TRAIN_PIPE_START = 'signal_train_pipe_start'
    AFTER_COMPUTE_LOSS = 'compute_loss'
    BEFORE_BACKWARD = 'signal_before_backward'
    TRAIN_PIPE_END = 'signal_train_pipe_end'
    SOLVE_END = 'signal_solve_end'

We implement the adaptive sampling method as follows.

class SpaceAdaptiveReceiver(sc.Receiver):
    # implement the abstract method in sc.Receiver
    def receive_notify(self, solver, message):
        # In each iteration, after the train pipe ends, the receiver will be notified.
        # Every five 500 iterations, the adaptive sampling will be triggerd.
        if sc.Signal.TRAIN_PIPE_END in message.keys() and solver.global_step % 1000 == 0:
            sc.logger.info('space adaptive sampling...')
            # Do extra sampling and compute the residual
            results = solver.infer_step({'data_evaluate': ['x', 't', 'sdf', 'AllenCahn_u']})
            residual_data = results['data_evaluate']['AllenCahn_u'].detach().cpu().numpy().ravel()
            # Sort the points by residual loss
            index = np.argsort(-1. * np.abs(residual_data))[:200]
            _points = {key: values[index].detach().cpu().numpy() for key, values in results['data_evaluate'].items()}
            _points.pop('AllenCahn_u')
            _points['area'] = np.zeros_like(_points['sdf']) + (1.0 / 200)
            # Update the points in the re_samping_domain
            solver.set_domain_parameter('re_sampling_domain', {'points': _points})

We also draw the result every $1000$ iterations.

class PostProcessReceiver(Receiver):
    def receive_notify(self, solver, message):
        if pinnnet.receivers.Signal.TRAIN_PIPE_END in message.keys() and solver.global_step % 1000 == 1:
            points = s.infer_step({'allen_test': ['x', 't', 'u']})
            triang_total = tri.Triangulation(points['allen_test']['t'].detach().cpu().numpy().ravel(),
                                             points['allen_test']['x'].detach().cpu().numpy().ravel(), )
            plt.tricontourf(triang_total, points['allen_test']['u'].detach().cpu().numpy().ravel(), 100)
            tc_bar = plt.colorbar()
            tc_bar.ax.tick_params(labelsize=12)
            plt.xlabel('$t$')
            plt.ylabel('$x$')
            plt.title('$u(x,t)$')
            plt.savefig(f'result_{solver.global_step}.png')
            plt.show()

Before Solver.solve() is called, register the two receivers to the solver:

s.register_receiver(SpaceAdaptiveReceiver())
s.register_receiver(PostProcessReceiver())

The training process is shown as follows:

See examples/allen_cahn.